linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

A

is the list of entries

a_

where

i = j

. All

off-diagonal element In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

s are

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

in a

diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...

. The following four matrices have their main diagonals indicated by red ones: :

\begin
\color & 0 & 0\\
0 & \color & 0\\
0 & 0 & \color\end
\qquad
\begin
\color & 0 & 0 & 0 \\
0 & \color & 0 & 0 \\
0 & 0 & \color & 0 \end
\qquad
\begin
\color & 0 & 0 \\
0 & \color & 0 \\
0 & 0 & \color \\
0 & 0 & 0 \end
\qquad
\begin
\color & 0 & 0 & 0 \\
0 & \color & 0 & 0 \\
0 & 0  &\color & 0 \\
0 & 0 & 0 & \color \end
\qquad

Antidiagonal

The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order

N

square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...

B

is the collection of entries

b_

such that

i + j = N+1

for all

1 \leq i, j \leq N

. That is, it runs from the top right corner to the bottom left corner. :

\begin
0 & 0 & \color\\
0 & \color & 0\\
\color & 0 & 0\end

References

* {{MathWorld, id=Diagonal, title=Main diagonal Matrices

Antidiagonal

See also

References